1. “Only,” Categorical Relationships, logical operators
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Review
Only vs. Only if
Categorical Relationships
The Square of Opposition
Logical Operators
Conversion
For Next time: Read Chapter 8 pages

2. Quick Review
We are now focusing on translating sentences into categorical claims
We have already had a lot of practice understanding the differences between the four different types of categorical claims:
A-Claim
E-Claim
I-Claim
O-Claim

3. Quick Review Some sentences are harder to translate than others
Indicator words like “most” or “many” or “a few” seem to be telling us different things about the relationships between different categories
For our purposes the following A-Claim is true: ALL of these words Are words that would be translated as _________?
If you see one of those words you know you are dealing with an I or O-Claim

4. Only
Things are a little more complicated with the indicator word “only”
“Only students may use the gym”
“Critical Thinking students are the only students who can avoid cognitive biases”
“Only” can indicate different relationships between groups depending on whether it is found by itself or with the word “the”
What is the word “only” saying in the two examples above?

5. Only (more)
Whenever we see “only” by itself in a sentence then we know we are being introduced to the predicate term in an A-claim
Only bananas are fed to the monkeys
Whenever we see “the only” in a sentence we know we are being introduced to the subject term in an A- claim
Red meat is the only thing fed to the lions

6. Example Franklin Delano Roosevelt once said:
“The only thing we have to fear is fear itself”
What are the subject and predicate terms of this sentence
How would we translate Roosevelt's sentence into a categorical claim?

7. Curious facts about Categorical Claim types
If we diagram the four claim types (A,E,I,O) we can notice that they are related to one another in different ways
It is impossible for both an A-Claim and an E-Claim to be true at the same time (about the same groups)
A-Claim: All Chihuahuas are cute
E-Claim: No Chihuahuas are cute
This makes A and E-Claims contraries. These two claim types cannot both be true together
What about I and O-Claims?

8. More Curious Facts I and O-Claims are related to one another as well
I and O-Claims can both be true together but they can never both be false
I-Claim: Some mammals are egg layers
O-Claim: Some mammals are not egg layers
This is because we will be working under the assumption that the groups we use in our claims are not empty
Because of this, I and O-Claims are called subcontrary claims

9. One More Curious Fact
Contrary claims cannot both be true but they can both be false
Subcontrary claims cannot both be false but they can both be true
But A and O-Claims AND E and I-Claims have a different relationship to one another: they never have the same truth value
If an A-Claim is true then the corresponding O-Claim must be false (and vice-versa)
For this reason A & O-Claims and E & I-Claims are called contradictory claims

10. The Square of Opposition

11. Inferences based on Categorical Claims
If we keep in mind the relationships between different kinds of categorical claims then we can infer the truth values of other claims once we know the truth value of one categorical claim
For example: if we know that the A-Claim “All squirrels are creatures with bushy tails” is true then we know that the E- Claim “No squirrels are creatures with bushy tails” is false
We know this because A and E-Claims are contraries, they cannot both be true
We also know that the I-Claim “Some squirrels are creatures with bushy tails” is also true

12. Inferences based on Categorical Claims
Similarly, we know that the two subcontrary claim types (I and O) can never both be false
If we know that the O-Claim “Some mammals are not cute” is false then we know that its subcontrary I-Claim “Some mammals are cute” must be true because they cannot both be false
The same goes for contradictory claims. If we know that the O-Claim “Some asparagus plants are not green things” is true then we know that the contradictory A- Claim “All asparagus plants are green things” is false

13. Quick Review
Knowing the truth value of one of the categorical claim types helps us to infer the truth values of some of the other claim types
A-Claim: if true we can infer that the E-Claim and O- Claim are false and the I-Claim is true
A-Claim: if false we can infer that the O-Claim is true but we can't say much more with certainty
The square of opposition can help us see those relationships and we will need them when we start doing proofs for validity

14. Practice
If we know that the A-claim “all circus clowns are scary things” is true, what can we infer about the truth values of the other categorical claims?
It might help to draw the Venn Diagram of the A- Claim to see what it rules out or rules in
What if we knew that “Some people are people who hate pizza”? What truth values could we infer from that claim?
Here again, it might help to draw the I-Claim diagram

15. Logical Operators
A logical operation is a way of translating a claim into another claim without changing the meaning or truth value of the claim
There is a corresponding sense of this process in mathematics
We can say that x + y = z is equivalent to y + x = z, for example, without changing the meaning of the expression or the truth value of the equation
In logic, being able to do these operations will be extremely useful when we construct proofs

16. Conversion The first equivalency we will look at is conversion
To find the converse of a claim all we have to do is switch the order of a claim's subject and predicate terms
Some claim types are equivalent to their converses. This means that the truth value and meaning of a claim is not changed by translating it into its converse
All E and I-Claims are equivalent to their converses
For example, diagram these two I-Claims:
Some elephants Are large animals
Some large animals Are elephants

17. Why Should We Care?
The important thing about logical operations like conversion is that we can always safely replace any I-Claim in an argument with its converse claim without changing the argument
The same goes for any E-Claim
For example:
No ghosts Are beings who exist
No beings who exist Are ghosts
When we check the Venn diagram we see that both of these two claims are equivalent as well

18. Complementary Terms
A complementary term refers to a group that excludes all members of another group
For example:
[Students] is a group; its complementary term would be a group that excludes every member of the [Students] group
We might call that group [Non-Students]
Every group in a categorical claim has a complementary group
[Bananas]/[Non-Bananas]; [Things that eat pizza]/[Things that do not eat pizza]; [boring things]/[non-boring things]

19. Obverse Claims
ALL categorical claims (A,E,I,O) are equivalent to their obverse claims
An obverse claim can be found by first looking at the claim to the left/right of a claim type on the Square of opposition
THEN switch the predicate term with its complementary term
Voila! This gives you the obverse claim

20. Examples For the following claims, find the obverse claim
All monkeys Are creatures who eat bananas
Some ants Are dangerous things
No circus clowns Are truly scary things
Here again we can check the Venn diagram to make sure that each claim and its obverse claim are equivalent

21. For Next Time
Read Chapter 8 pages